Assuming we need to find the number in each cell, here’s a variant that will give the others a bit more information: “is the total either 9 or 21?”
this shows the others that Kleene has 4, 5, or 6. I don’t see that that’s enough yet. But maybe y’all do.

Do we all agree on what the task really is?
is it (1) ensure someone can correctly state the total number of coins —seems like that is solved by Molly Mae, OR (2) ensure someone can correctly identify the number of coins in each cell?

Well, there goes your curve estimation. I don't remember the situation but there were certain combinations that I could calculate with some multiplication. I think one of them was a CE combination. But of course the other combinations didn't feel like cooperating. I'll try to work on this over the weekend.
Agree on your pink combinations except for XMEE. Should be the same count as EEM, right?

@Pickett Interesting result. OP meant for the coins to have equal value, but yours is a great puzzle also, even for finite numbers of coins received.
@rocdocmac and @Pickett When midnight has struck an infinite number of events will have transpired and the process will have stopped. Will any of the three be happier than the others?

At an ever increasing pace Al, Bert and Charlie have been receiving into three identical boxes of limitless capacity identical pairs of silver coins engraved with the integers 1, 2, 3, 4, ... etc. These events occur on a precise schedule, each box receiving
Coins marked 1 and 2 at 1 minute before midnight
Coins marked 3 and 4 at 1/2 minute before midnight
Coins marked 5 and 6 at 1/4 minute before midnight
Coins marked 7 and 8 at 1/8 minute before midnight
etc.
But they were instructed at each event to remove a coin from their respective boxes and discard it. After some thought, Al decided each time to discard his lowest-numbered coin; Bert discarded an even-numbered coin; and Charlie thought what the heck and discarded a coin selected at random. Regardless of strategy, at each event the number of coins in each box grew by unity, so that after N events each box held N coins.
Needless to say when midnight struck their arms were infinitely tired, but it was a small price to pay for infinite riches. But tell us, now, whether their expectations were met.
Describe the contents of each box at midnight.